Pedestrian routing fails around Bocchetta dei Tre Signori
openrouteservice and graphhopper also both fail similarly.
Osmand works
My only suspect is sac_scale=demanding_alpine_hiking but I was not able to find another similarly tagged way to test my idea.

Yes, you are correct. The two highest grades, T5 and T6, are excluded from automatic routing as they are described as "needing alpine equipment".

The rationale is that when planning tours on such difficulty levels, you should know what you're doing and not rely on automatic routing. On the other hand however, sending a regular hiker up there unintentionally might lead to life-threatening situations. I really wouldn't want to be responsible for such a thing happening.